Calculator
Example Data Table
| Triangle Type | Known Side | Known Value | Computed Sides | Exact Relation |
|---|---|---|---|---|
| 45-45-90 | Leg | 8 | 8, 8, 11.3137 | 8, 8, 8√2 |
| 45-45-90 | Hypotenuse | 10 | 7.0711, 7.0711, 10 | 5√2, 5√2, 10 |
| 30-60-90 | Short Leg | 6 | 6, 10.3923, 12 | 6, 6√3, 12 |
| 30-60-90 | Hypotenuse | 18 | 9, 15.5885, 18 | 9, 9√3, 18 |
Formula Used
Special right triangles follow fixed angle patterns and constant side ratios. That makes unknown sides predictable from one known side.
45-45-90 Triangle
- Angles: 45°, 45°, 90°
- Side ratio: 1 : 1 : √2
- Hypotenuse = Leg × √2
- Leg = Hypotenuse / √2 = Hypotenuse × √2 / 2
30-60-90 Triangle
- Angles: 30°, 60°, 90°
- Side ratio: 1 : √3 : 2
- Long Leg = Short Leg × √3
- Hypotenuse = 2 × Short Leg
- Short Leg = Long Leg / √3 = Long Leg × √3 / 3
- Long Leg = Hypotenuse × √3 / 2
Derived Measures
- Perimeter = a + b + c
- Area = (a × b) / 2
- Semiperimeter = (a + b + c) / 2
- Inradius = Area / Semiperimeter
- Circumradius = Hypotenuse / 2
- Altitude to Hypotenuse = (a × b) / c
How to Use This Calculator
- Select the special triangle type.
- Choose which side value you already know.
- Enter the side length.
- Optionally enter a unit label and decimal precision.
- Press the calculate button.
- Review exact forms, decimal outputs, and derived measures.
- Use the CSV or PDF buttons to save the result.
- Inspect the graph to confirm the triangle shape visually.
Frequently Asked Questions
1. What is a special right angle triangle?
A special right angle triangle has a fixed angle pattern and a fixed side ratio. The two standard cases are 45-45-90 and 30-60-90 triangles.
2. Why are these triangles useful?
They simplify geometry, trigonometry, construction, and exam problems. One side immediately determines the others through exact ratios, which saves time and reduces algebra.
3. Can I enter the hypotenuse directly?
Yes. The calculator accepts the hypotenuse for both supported triangle types. It then converts that value into the correct leg lengths and other measurements.
4. What does the exact form mean?
Exact form keeps radicals like √2 or √3 instead of rounding them. This is useful for proofs, symbolic work, and homework that requires exact answers.
5. Does the calculator also find area and perimeter?
Yes. It computes side lengths, perimeter, area, semiperimeter, inradius, circumradius, and the altitude from the right angle to the hypotenuse.
6. What unit should I enter?
You can enter any label such as cm, m, ft, or in. The calculator uses that label in the displayed results and exported files.
7. Why does the graph matter?
The graph helps verify orientation and relative side lengths. It is useful for checking whether the computed triangle matches your expectations visually.
8. Is this calculator suitable for study and practical work?
Yes. It is suitable for classwork, homework, design checks, layout planning, and quick verification whenever a special right triangle appears.